Published in Science China Mathematics
We give a classification of second-order polynomial solutions for the homogeneous k-Hessian equation σk[u] = 0. There are only two classes of polynomial solutions: One is convex polynomial; another one must not be (k + 1)-convex, and in the second case, the k-Hessian equations are uniformly elliptic with respect to that solution. Base on this class...
Published in Science China Mathematics
We study the quasisymmetric geometry of the Julia sets of McMullen maps fλ(z) = zm + λ/zℓ, where λ ∈ ℂ {0} and ℓ and m are positive integers satisfying 1/ℓ+1/m
Published in Science China Mathematics
As usual, denote by P(n) the largest prime factor of the integer n ⩾ 1 with the convention P(1) = 1. For 0
Published in Science China Mathematics
In this paper, we establish a quite general mean value result of arithmetic functions over short intervals with the Selberg-Delange method and give some applications. In particular, we generalize Selberg’s result on the distribution of integers with a given number of prime factors and Deshouillers-Dress-Tenenbaum’s arcsin law on divisors to the sho...
Published in Science China Mathematics
We use the Weyl correspondence approach to investigate the spectral operators of the Laguerre and the Kautz systems. We show that the basic functions in the Laguerre and the Kautz systems are the eigenvectors of modified Sturm-Liouville operators. The results are further generalized to multiple parameters of one complex variable in both the unit di...
Published in Science China Mathematics
Based on the distortion theory developed by Cui and Tan (2015), we prove the landing of every parameter ray at critical portraits coming from non-recurrent polynomials, thereby generalizing a result of Kiwi (2005).
Published in Science China Mathematics
For equations of order two with the Dirichlet boundary condition, as the Laplace problem, the Stokes and the Navier-Stokes systems, perforated domains were only studied when the distance between the holes dε is equal to or much larger than the size of the holes ε. Such a diluted porous medium is interesting because it contains some cases where we h...
Published in Science China Mathematics
In this survey we report some recent results on the dynamics of a rigid body immersed in a two-dimensional incompressible perfect fluid, with an emphasis on the zero-radius limit.
Published in Science China Mathematics
In this paper, we develop semi-classical analysis on H-type groups. We define semi-classical pseudodi fferential operators, prove the boundedness of their action on square integrable functions and develop a symbolic calculus. Then, we define the semi-classical measures of bounded families of square integrable functions which consist of a pair forme...
Published in Science China Mathematics
In this short note we revisit the convex integration approach to constructing very weak solutions to the 2D Monge-Ampére equation with Hölder-continuous first derivatives of exponent β
